7th Asia-Pacific Conference on Plasma Physics, 12-17 Nov, 2023 at Port Messe Nagoya

Clebsch representation and generalized enstrophy for relativistic plasma
1K. Nunotani1 and Z. Yoshida1,2
Graduate School of Frontier Sciences, The University of Tokyo
2
National Institute for Fusion Sciencee-mail (speaker): nunotani.keiichiro20@ae.k.u-tokyo.ac.jp

𝑑The theory of relativistic plasmas is useful to model
𝑄 (𝑡) = ∫ { 𝑓 ′ (𝜗) 𝑐 𝐽 (log 𝛾 , 𝜆𝑘 , 𝜎𝑘 , 𝜎3−𝑘 )high-energy astronomical objects. In this context,
𝑑𝑡 𝑘Ω(𝑡)topological constraints built in the governing equations−𝑓(𝜗) 𝑛 𝑢𝜇 𝜕𝜇 (log 𝛾) } 𝑑3 𝑥 (5)play an essential role in characterizing structures selfwhere 𝐽 (𝜉 0 , 𝜉 1 , 𝜉 2 , 𝜉 3 ) is the Jacobian determinant.
organized by plasmas. Among various invariants of ideal
Here 𝑑𝑄𝑘 /𝑑𝑡 is proportional to the derivative of log 𝛾,models, the circulation is one of the most fundamental
quantities, being included in other invariants like theimplying that 𝑄𝑘 (𝑡) is not preserved in a relativistic
helicity. The conventional enstrophy in a twosetting. Finally, we derive a relativistic enstrophy 𝔔𝑘 (𝑠)dimensional (2D) flow is defined asthat is preserved by the relativistic fluid equations. An𝜔2
explanation of the geometric meaning carried by 𝔔𝑘 (𝑠)(1)∫ 𝑓 ( ) 𝜌 𝑑𝑥
𝜌is given in figure 1. We also discuss simple examples andΣ(𝑡)show that highly symmetric flows may preserve bothwith a particle number density 𝜌, the scalar vorticity 𝜔,
𝑄𝑘 (𝑡) and 𝔔𝑘 (𝑠) due to the vanishing of 𝑢𝜇 𝜕𝜇 (log 𝛾).
an arbitrary smooth function 𝑓: ℝ → ℝ, and an arbitrary
This work was supported by JSPS KAKENHI Grantregion Σ(𝑡) ⊂ ℝ2 that co-moves with the 2D fluid.
No. 17H01177 and No. 23KJ0435.
Enstrophy (1) is known to be constant and to measure thenumber of vortex filaments penetrating the surface Σ(𝑡).
References
By introducing potential fields 𝜑, 𝜆1 , 𝜎1 , 𝜆2 , and[1] P. J. Morrison, Rev. Mod. Phys. 70, 467 (1998).
𝜎2 , the velocity field 𝑉 of a fluid can represented as[2] Z. Yoshida, J. Math. Phys. 50, 113101 (2009).
(2)𝑚𝑉 ∶= ∇𝜑 + 𝜆1 ∇𝜎1 + 𝜆2 ∇𝜎2 .
[3] Z. Yoshida and P. J. Morrison, in Mathematics forThis expression is called a Clebsch representation of 𝑉,
nonlinear phenomena — analysis and computation,and the potential fields above are called Clebsch
edited by Y. Maekawa and S. Jimbo (2017), pp. 271–284.
parameters or Clebsch variables [1]. Mathematically, a[4] Z. Yoshida and P. J. Morrison, Phys. Rev. Lett. 119,
general three-dimensional (3D) vector field 𝑉 can
244501 (2017).
always be cast into the form of equation (2) [2]. TheK. Nunotani and Z. Yoshida, Phys.
29,
Physics of[5]PlasmasARTICLE Plasma
scitation.org/journal/php
corresponding vorticity vector 𝜔 can be written as052905 (2022).
(3)
𝜔 = ∇𝜆1 × ∇𝜎1 + ∇𝜆2 × ∇𝜎2 .
By invoking Clebsch variables, the conventional
enstrophy in a 2D flow can be generalized to a 3D flow:
for 𝑘 = 1, 2,
𝜔 ⋅ ∇𝜎3−𝑘𝑄̃𝑘 (𝑡) ≔ ∫ 𝑓(𝜗𝑘 ) 𝜌 𝑑𝑥3 ,𝜗𝑘 ≔ 𝑘(4)𝜌Ω(𝑡)with 𝜔𝑘 ≔ ∇𝜆𝑘 × ∇𝜎𝑘 . The integration domain
Ω(𝑡) ⊂ ℝ3 is an arbitrary region that co-moves with the3D fluid. Both 𝑄̃1 (𝑡) and 𝑄̃2 (𝑡) can be interpreted as
topological charges of a 3D fluid element, which
essentially measure circulation [3,4].
FIG. 1. A conceptual
drawing of t-planeand s-plane(right) of the fluid flowingdrawing
outward radially from theThe t-plane is a hyperplane
Figure1.(left) Aconceptual
oforigin.𝑡-plane(left)in space-timeandat constant time t
Since relativistic effects impart space-time coupling
and used for the integral domain of the semi-relativistic enstrophy Q(t). On the other hand, the s-plane is a curved hypersurface in space-time reflecting the intrinsic time s ofthe fluid and used for the integral domain of the relativistic enstrophy QðsÞ. In this figure, the fluid farther from the origin is faster, so more time t has passed on the outer side.
into the metric, such invariants are no longer constant in𝑠-plane (right) of the fluid flowing outward radially from
Note that in the definition of #, T ðsÞ ðx ! dr Þ and T ðsÞ ði nÞ
We chooseto compare
the two The
3-forms x𝑡-plane! dr and i n.isThea hyperplanea relativistic setting. In particular, the non-relativistic
theorigin.
in space-time atare three-forms on a three-dimensional manifold V & R , which
linear space X ðMÞ is four-dimensional, and x ! dr and i n are not
become 0-forms if we let Hodge star # act on them. Since
necessarilyparallel.
In
ordertocomparethem,
weuse
acoordinate
enstrophy is no longer conserved in a relativistic plasma,constant
time𝑡andusedfor
theintegral
domain of themap on V(s). We hope that once the two 3-forms are pulled back, one(51)L ði nÞ ¼ L ðx ! dr Þ ¼ 0of the other as three-forms on a 3D submanifold. A
implying that the conservation of circulation is violated.is a scalar multiple
semi-relativistic
enstrophy𝑄(𝑡).
Ontheotherhand, thenatural coordinate map on V(s) seems to be the diffeomorphism
1
can be derived from the time evolution of Clebsch parameters fromT ðsÞ : V ! VðsÞ.
In this work [5], we extend the (non-relativistic)
Eqs. (42a) and (42b),
is a curved
hypersurfacein space-time reflecting
We must 𝑠-planeverify that the derivativeby s of the pullbackby T ðsÞddenstrophy in a 3D flow to a Lorentz covariant form. To
and the Lie derivative L are connected. For a scalar function
T ðsÞ ði nÞ ¼ T ðsÞ ðx ! dr Þ ¼ 0(52)theintrinsic
time𝑠ofthefluid
anddsused for dsthe integral
g 2 C ðMÞ,!
"this end, we first derive relativistic fluid equations for thedd@g d
can be obtained from Eq. (48). Therefore, the conservation law of theTdomainT ðsÞðsÞ g ¼ g ofT ðsÞ¼ relativistic
theenstrophy
𝔔1 (𝑠).
In this figure,relativistic enstrophyQðsÞ follows, that is,ds@x ds
Clebsch parameters by using the principle of least action. ds@g
thefluidtime 𝑡¼
U jfarther¼ i dgj from the origin is faster, sod moreQðsÞ ¼ 0:
(53)
@x
We then show that the time evolution of the conventionalds¼ L gj¼onT ðsÞtheðL gÞ:(47) [5].
has passed
outer sideThedefinitionof
QðsÞ
inEqs.
(49)
and(50) can be written symenstrophy 𝑄𝑘 (𝑡) is given by:
Any k-form can be written as a linear sum of exterior derivatives ofbolically as follows:
13

1

2#UU

212U0UUU#U1U

3#

U

20UUU

1UU#

VðsÞf ð#Þ iU n;

12l

U#!

Ull

UT U ðsÞUlUT U ðsÞU#T U ðsÞUscalar functions combined with wedge products. Since a pullback iscommutative with a wedge product, and both dsd and LU satisfy theLeibniz rule and are commutative with an exterior derivative, Eq. (47)holds for general k-forms (not only for 0-forms)
dT U ðsÞ# x ¼ T U ðsÞ# ðLU xÞ;(48)
dskfor any x 2 X ðMÞ. Based on this fact, the relativistic enstrophy QðsÞis defined as follows:ð(49)QðsÞ :¼ c%1 f ð#Þ T U ðsÞ# ðiU nÞ;
V0with a scalar function# :¼ c$%#T U ðsÞ# ðx1 ! dr2 Þ$%# :T U ðsÞ# ðiU nÞ(50)QðsÞ :¼ c%1# :¼ cð

x1 ! dr2:iU n(54)(55)

This simple formalism would make it easier to understand the definition and to compare it with conventional enstrophy (22). However,this is only symbolic and ill-defined, and the actual calculation is asshown in Eqs. (49) and (50).
In the following, we will check the correspondence of the semirelativistic generalized enstrophy and the relativistic generalized enstrophy in the non-relativistic limit. This is one of the reasons for thevalidity of the definition of the relativistic generalized enstrophy. It issufficient to verify that the semi-relativistic enstrophy is identical with